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One of the things I’d like to use more often in my Precalculus classes are multiple choice questions to promote class discussion. Today as part of the opener I gave my first MC question. Fairly straight forward question because I wanted to get the kiddos logged into the Nspire Navigator, which we haven’t done for a while. The students wrote down on their Opener-Exit slip the answer along with any calculations they needed to do to determine the best choice. Then the Navigator, through a Quick Poll, collects their answers. Once I stop the poll, I can project the responses in a bar chart.

As you can see, there were two popular answers. I asked my students in their groups to discuss their choices and try to determine which one was actually the correct answer along with supplying a reasoned argument for their group’s choice. Some great conversations! I then resent the Quick Poll to see if they could arrive at the correct answer without my confirming the answer.

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One activity I started last year to get students thinking about constraints and possibilities of an application problem is what I call the “Folder Activity.” This time around, I wanted my students to be thinking about what makes a problem sinusoidal and what information from the situation helps to create a mathematical model (equation). In addition, I want them to be thinking about what kinds of questions could be asked about the situation that can be answered with the graph and/or the function equation.

To set up this activity, I needed lots of sinusoidal application situations that are different from the typical situations of ferris wheels, merry-go-rounds and oscillating springs. The internet is a great source for these and I found many! I then “pull apart” the problem situation from the questions asked. The students only get the problem situation…here is one example of the eight different problems I had available:

First of all, students open the task card on their iPad. This includes group roles, materials needed, the task and the product. Here is a copy of the tasks and folder layout.

The groups then get a folder, a quarter sheet headed with “Known Information,” a second quarter sheet headed with “Mathematical Relationships,” and a half sheet of graph paper. At this point they read and discuss the problem information, determine known information and identify potential useful mathematical relationships…they know not all of the possible mathematical relationships will be used, but it is helpful to think about them. The discussions are rich, the misconceptions get “fixed” most of the time, and once the questions are actually asked, they already have a plan to answer them.

Once the information is organized and glued to the folder, then the group can pick up the actual questions and begin working on them.

The final product is the folder with the solution written out neatly…each person’s handwriting needs to be apparent in the solution write-up. All is glued in the proper places and finally the group evaluates to what extent they used the 8 Math Practices along with a 1-2 sentence summary for each practice.

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I’ve been reading the book, Total Participation Techniques: Making Every Student an Active Learner by Persida Himmele and William Himmele, published by ASCD (can also be found online and downloaded if you are a member of ASCD). The authors talk about getting your students actively engaged AND cognitively invested in the learning rather than being “listening objects” in the classroom. One quote that resonated with me was:

At any age, people need to pause and process what they are learning. They need to chew on concepts, jot down their thoughts, compare understandings with peers, and articulate their questions…and celebrate the learning that is happening right now in my head.

As a follow-up to solving trig equations, I had my students create posters, but I wanted them to do more. I found a great idea from Rebecca Peterson called the Mistakes Game. I adjusted the directions some and came up with this activity:

The Tasks:

SOLVE CORRECTLY:

As a group, work your given problem correctly. Then, check your answer using graphing technology.

Once you have the correct answer(s), write out the solution process neatly on a poster paper.

Include a graph of the original function and the location of the solutions (color-code by principle solution and symmetry solution.)

INTRODUCE A COMMON MISTAKE:

Now work the problem incorrectly, hiding your mistake as cleverly as possible. Your “mistake” must be a true pitfall of the given problem (i.e., what kinds of conceptual errors would students likely make?). Your error cannot be a simple arithmetic or algebraic mistake unless it is related to using an Algebra Trick incorrectly.

When you’re happy with your “lie”, put it on the back of your poster paper. Post this side for all to see.

FIND THE ERRORS:

When every group is done, you will find the errors on the other posters and vote on the group with the sneakiest mistake.

Be prepared to discuss/defend your results.

I really like the idea of coming up with a mistake. This causes the students to use higher-order thinking skills to analyze and synthesize. This is cognitive engagement at its best. I heard lots of discussion around the process, with students correcting others and helping them understand their error; but then the magic happened! Trying to think of a mistake one could make when solving a trig problem, and the idea that they need theirs to be “tricky” led them to go through many of the common errors. They were talking about what would be an error and by deciding if it was “tricky” enough, they had to understand the nature of the error. I am really excited to see if this process helps students to think more carefully while they solve trig equations and to avoid those common pitfalls.

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In precalculus, we have begun to look at solving trigonometric equations. Yesterday, we developed the general rules for finding solutions for the Big Three trig functions: sin-1x + 2πn, π – sin-1x + 2πn, cos-1x + 2πn, 0 – cos-1x + 2πn, and tan-1x + πn

I am a bit of a stickler as I require students to use the inverse trig function definitions correctly while solving equations. As an opener, students were asked to do the following:

Find all solutions for the equation: sin2x = – ½. Now find the particular solutions in the domain: [1,5]. Verify solutions graphically.

My goal for this problem was to connect the algebraic process of solving the equation to the graphical results; that is, I wanted multiple representations and understanding for what was really happening.. Once again I wanted the conceptual underpinning to be solid as my kiddos practiced equation solving procedures; that is, I didn’t want mindless robots solving equations without thinking about what was really happening.

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We are finally at the end of the introduction to trigonometry in precalculus…and to have an assessment that covers the breadth of the topics, my teaching partner and I decided we would create a two-day assessment, one day being non-calculator part and the other day being a calculator part; so each day is written as a class period assessment. In order to get ready, my students were given a review sheet, but in class, having me drone on about the topics and going over problems can be deadly. So….we created 8 Trig Stations (see the post from last year to get some more details) that cover the 8 big topics. Students are in groups of 3-4 and have 10 minutes to work at each station…even if they don’t finish, they move on. This year, students requested to take photos of the questions if they didn’t finish along with the answers (found in the orange envelopes).

Most students found the experience helpful for identifying the areas they needed to focus on for studying. This year I even taped some of the conversations (unbeknownst to the group) so I could hear them use some of the mathematical practices including Sense-make, Reason, Argument, Model, Tools, Precision of language use, and Structure. They were very enlightening and encouraging!

What do you do to check on student use of the math practices?

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I asked my precalculus students this, “what do you think happens when you add, multiple or compose trigonometric functions with functions from other families? Will the combined functions be periodic? What type of shape might you expect?” I had four explorations ready for them to test out conjectures about what happens. I prefaced the experience by encouraging them to ask “What if? “How?” “Why?” questions as they worked through the explorations…and some actually did.

Initial student comments around the beginnings of the explorations: “It’s a sine function on a slant!” “How are we ever going to find the equation?” “Oh slam…you’re right!” It was so engaging for me as the teacher listening to the conversations, the conjecturing, and the testing/revising until my kiddos found an equation that actually did what they wanted. They also looked at predicting what an equation would look like graphed before they graphed it. Lots of dendrites growing today!

Here is one that combined sine, cosine and tangent. When asked how each showed up in the final graph, my students were able to point out exactly what they saw and why.

Finally, I had one student on his computer using Desmos with the exploration. And its available on the iPads for free. Although I am a big fan of the TI Nspires, I do wonder if Desmos might be a good alternative for some settings. Have to ruminate on that!

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Today I wanted my precalculus students to recap the coffee filter activity. But how to do it in an interesting way?! I found a great website site that has some awesome .gif files of mathematical concepts in picture form. Check it out!! There are some pretty amazing visual depictions of important (or not so important) math ideas.

Here is the one I found for visually communicating the definition of a radian. Love it!!

So the Opener question I asked was this:

Explain how this .gif demonstrated the definition of a radian. What is the conversion relationship between radians and degrees?

Also, as an added a reminder of the mathematical practices, I asked my kiddos which math practices they used to complete their write-up. Lots of bang for the time allotted! I think I will continue to use this as a follow-up to the “What in the World is a Radian?” activity.

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I love multiple choice and true/false questions as vehicles for authentic student argumentation as per the 3rd CCSS Mathematical Practice (highlights are mine):

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Good MC questions are hard to come by, but even some of the questions from our book create that focal point for students to build an argument, present it to their peers, and critique the reasoning of others. On today’s worksheet, I inserted 8 MC and TF questions along with a GradeCam blank form for students to record their final choices. This gives students a chance to listen to their peers arguments, but have the ultimate decision to accept the choice or go with their own argument.

Once students decide on their choices, they come up to my computer and have their iPad scanned in student mode to see how they did. GradeCam in student mode identifies the problem number of incorrect choices, but does not tell them the correct answer…also keeps it anonymous from other students so the “fear” of public mistakes is alleviated. Students know which questions to go back to and are eager to “fix” their thinking.

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Students get scared of long-winded problems. For example, this problem doesn’t even have the questions yet, but they freak out!

I used the same activity about the Crop Duster as last year to start out this block class activity. The premise is to get students to think about the situation without initially knowing what the questions about the situations are. Once they organize the information into “What is known?” “What are the possible mathematical relationships?” and “Sketch the situation” students are then given the actual questions to solve.

In using the follow-up activity, I’ve found that the hardest thing for students to determine are the potential mathematical relationships. Once they have the question, they totally forget these relationships and almost blindly try to solve without thinking about them. This activity slows them down and gets them really thinking about the mathematical constructs in the problem. They also learn that all potential mathematical relationships may not be used, but recognizing them helped clarify how to approach the questions.

I hope to do this activity with different function types throughout the year so that I have some student work examples for a presentation I hope to give at the Texas Instruments International Conference in Orlando Florida next February. Here is my submission:

Using a Problem-Solving Activity To Develop Mathematical Habits of Mind

Have you ever wondered how to help your students to think thoughtfully about a non-routine problem situation before diving in to solve it? Or help them persevere during the needed productive struggle phase? Or encourage students to use meta-cognition during and after the problem-solving process? So did I!! This student-centered problem-based collaborative learning activity requires students to read the problem thoughtfully and then obliges them to work and think together to organize what they know (including a graphical representation) generate questions determine an answer and finally communicate the solution in a cohesive and understandable way. Come and enjoy the fun!

I’m a little nervous at this point, but I know as the year progresses I’ll have more student work to share and some suggestions for how to use in other classes besides precalculus.

Its the day before Spring Break. We finished the Midterm yesterday and they did brilliantly! Not a great time to introduce new material. An easy way out is to “show a movie” but to waste precious time with students with an experience that has no real academic purpose goes against everything I believe.

So, how do I keep my precalculus students engaged and interested? I wanted something that reviewed an old, but not critical, learning but added an intellectual twist that will catch and keep their attention. And I remembered a workshop I attended about 5 years ago (I’m sorry but I just can’t find the name of who presented and would love to give you credit…just comment below with your name and any other info you’d like to share) using the Nspire to look at the relationship between polar and rectangular equations in graph form. The presenter shared 3 Nspire documents that had nice animations.

For this experience, though, I used Geogebra because the kids could manipulate easily with their iPads.

It was the 4th document which inspired me to create an investigation around a system of two polar equations. Luckily the document was created, but I wanted my students to reflect more while using the Nspire document.

For instance, I had the students do this:

Move to 2.1. Without talking to your group, watch the animation alone! You must watch carefully because you cannot re-graph again. Try to observe how many times these graphs’ paths intersect.

What did YOU notice?

Once everyone has completed the task alone, talk with your group. What did other people in your group notice? Try to paraphrase your discussion.

Then later the real thinking started. My students used their Nspire to do the following:

Move to 4.1. You will see the results of graphing the two polar graphs in the rectangular plane. Does this affect your answer? Be specific!

Use the rectangular graphs to give the coordinates of the points (in polar form please) of intersection of the limacon, r1 = 3 + 2 cosθ, and the four-leaved rose, r2 = 5 sin(2θ). Label the intersection points on both the rectangular graph and the polar graph. The first point is labeled P1 on both graphs.

Why could the apparent intersection point Q1 be called a “false” intersection point? What aspect(s) of a polar graph make it appear to be a point of intersection? Label the other false intersection points on the polar graphs in the same manner.

Show on the rectangular graphs above that the second-quadrant angle θ for point Q1 corresponds to a point on the limacon but not to a point on the rose. What are the coordinates of point on the rose that correspond to the location Q1? Is there a mathematical relationship between the point on the limacon related and the point on the rose which correspond to the location Q1?

I was so delighted with how engaged my students were…to see their heads down, fingers posed over their iPads and some conversations, too. The Math Practices were everywhere today! In fact, they didn’t even realize class was over, and that’s a biggie for a 7th period just before Spring Break.